If a person insisted that I trust them at their word, and/or acted offended when I asked for permission to run a background check, that would raise a red flag for me. I can tell right away that the person is trying to hide something. If they have no criminal record, and their credit is good, then there is no reason to object. If they are trying to hide something, then they have something to hide.
Besides, for every candidate that refuses to allow a background check, there are many more who don't mind at all.
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The mere fact that science may not yet adequately explain an object, event, or experience does not mean the immediate explanation should automatically default to a conspiratorial, extraterrestrial, paranormal, or supernatural cause.